bookshelf/Bookshelf/Enderton/Set/Chapter_6.lean

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import Mathlib.Data.Set.Function
import Mathlib.Data.Rel
/-! # Enderton.Set.Chapter_6
Cardinal Numbers and the Axiom of Choice
-/
namespace Enderton.Set.Chapter_6
/-! #### Theorem 6A
For any sets `A`, `B`, and `C`,
(a) `A ≈ A`.
(b) If `A ≈ B`, then `B ≈ A`.
(c) If `A ≈ B` and `B ≈ C`, then `A ≈ C`.
-/
theorem theorem_6a_a (A : Set α)
: ∃ f, Set.BijOn f A A := by
refine ⟨fun x => x, ?_⟩
unfold Set.BijOn Set.MapsTo Set.InjOn Set.SurjOn
simp only [imp_self, implies_true, Set.image_id', true_and]
exact Eq.subset rfl
theorem theorem_6a_b [Nonempty α] (A : Set α) (B : Set β)
(f : α → β) (hf : Set.BijOn f A B)
: ∃ g, Set.BijOn g B A := by
refine ⟨Function.invFunOn f A, ?_⟩
exact (Set.bijOn_comm $ Set.BijOn.invOn_invFunOn hf).mpr hf
theorem theorem_6a_c (A : Set α) (B : Set β) (C : Set γ)
(f : α → β) (hf : Set.BijOn f A B)
(g : β → γ) (hg : Set.BijOn g B C)
: ∃ h, Set.BijOn h A C := by
exact ⟨g ∘ f, Set.BijOn.comp hg hf⟩
end Enderton.Set.Chapter_6